Apply Snell geometry without inventing impossible rays

1. Define the evidence and frame

Body-wave paths bend when wave speed changes. For a plane isotropic interface, the ratio sin(angle from normal)/speed is conserved across a transmitted ray. A faster lower layer therefore bends the ray away from the normal, not toward it. Angles measured from the horizontal interface would use a different geometric expression.

2. Connect mechanism and model

Here the upper speed is 6 km/s and the lower speed is 6, 8 or 10 km/s. The candidate transmitted sine is (v2/6)sin i. If it exceeds one, there is no real transmitted body-ray angle of this simple type. The interface can still support critical-refraction behavior and other wave phenomena; the model does not assert that all energy disappears.

The critical incidence angle for v2 greater than 6 is arcsin(6/v2), at which the transmitted direction becomes parallel to the interface. At equal speeds the formal limit is 90 degrees and there is no bending. Do not silently clamp an impossible sine to one and present that as a measured horizontal ray.

Global seismic interpretation combines many phases, refraction paths and travel times through a curved, layered Earth. This two-layer geometric exercise isolates one mechanism. It does not identify a core boundary or uniquely recover interior structure from a single angle.

3. Reconstruct the worked packet

At incidence 30° and lower speed 8 km/s, sin r=2/3, so r≈41.8103°. The critical incidence is arcsin(6/8)≈48.5904°. A 60° incident ray would require a sine greater than one.

4. Predict and investigate

Predict first: At a faster lower layer, does the transmitted angle become larger or smaller relative to the normal?

Change incidence across the critical threshold and retain the sine test before attempting an inverse sine. Sketch both directions and the interface normal. Record the no-propagating-ray condition rather than filling it with a fabricated angle.

5. Test a changed case

Apply the evidence to a changed case

For incidence 60° and lower speed 10 km/s, is a propagating transmitted body ray admitted by this model?

Retain the evidence frame, calculations and assumptions before opening public feedback.

Compare a worked response

No. The required sine is (10/6)sin60°≈1.4434, outside the real-angle domain. The critical angle is about 36.8699°; additional interface-wave physics is outside this calculation.

6. Retain a defensible evidence notebook

Preserve first predictions, full units, geometry, source provenance, assumptions and rejected alternatives. Synthetic models are not field measurements. The explicitly labeled USGS packet is actual catalog evidence, but working through a public example is not independent practical competence or an operational hazard assessment.

Mechanism and data reference, checked 30 October 2026. Original teaching prose, diagrams and exercises; actual USGS rows retain their agency and provenance.